I like the answer from s6k, and I think it answers Maryrrf's original question. Mary, if you can hear the difference between major and minor chords and hear songs which are primarily major or minor, you're on you way to "hearing" what I think you mean by the "modal" sound. I personally like the "Greensleeves" or "What Shall We Do With a Drunken Sailor" dorian sound.

In your original question, "What would tip you off to the fact that it is modal?"...we've talked about why that is not really a "proper" question, but I think I understand what you mean. My answer would be...if it sounds "different" from major or minor, then it's one of the other modes. (and we often slangily refer to the "other ones" as modal.)